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List of functions
This page lists various googological functions arranged by growth rate according to the fast-growing hierarchy. *\(\approx\) means that two functions have comparable growth rates. *\(>\) means that one function significantly overgrows the other. *\(\geq\) means that it is not known exactly whether one function overgrows the other or not. *(limit) means that the function has many arguments, and the growth rate is found by diagonalizing over them. From \(f_0(n)\) to \(f_\omega(n)\) These functions are primitive recursive. *Addition \( = a+b > f_0(n)\) *Multiplication \(= a*b > f_1(n)\) *Exponentiation \(= a^b \approx f_2(n)\) *Factorial (and most of its extensions) \(= n! \approx f_2(n)\) *Exponential factorial \(\approx f_3(n)\) *Tetration \(= {^{b}a} \approx f_3(n)\) *Pentation \(= a \uparrow\uparrow\uparrow b \approx f_4(n)\) *Circle notation \(\approx f_4(n)\) *Hexation \(= a \uparrow^{4} b \approx f_5(n)\) *Heptation \(= a \uparrow^{5} b \approx f_6(n)\) *Octation \(= a \uparrow^{6} b \approx f_7(n)\) *Enneation \(= a \uparrow^{7} b \approx f_8(n)\) *Decation \(= a \uparrow^{8} b \approx f_9(n)\) *Undecation \(= a \uparrow^{9} b \approx f_{10}(n)\) *Doedecation \(= a \uparrow^{10} b \approx f_{11}(n)\) *Tredecation \(= a \uparrow^{11} b \approx f_{12}(n)\) From \(f_\omega(n)\) to \(f_{\omega^\omega}(n)\) *Weak goodstein function \(= g(n) \approx f_\omega(n)\) *Ackermann function \(= A(n,n) \approx f_\omega(n)\) *Ackermann numbers \(\approx f_\omega(n)\), the limit of the hyper operators in general *Steinhaus-Moser notation \(\approx f_\omega(n)\) *Arrow notation (both variants) \(= a \uparrow^{c} b \approx f_\omega(n)\) *Hyper-E notation \(= E\# \approx f_\omega(n)\) (limit) *Graham's function \(= g_n \approx f_{\omega+1}(n)\) *Expansion \(= a \{\{1\}\} b \approx f_{\omega+1}(n)\) *Multiexpansion \(= a \{\{2\}\} b \approx f_{\omega+2}(n)\) *Powerexpansion \(= a \{\{3\}\} b \approx f_{\omega+3}(n)\) *Expandotetration \(= a \{\{4\}\} b \approx f_{\omega+4}(n)\) *Explosion \(= a \{\{\{1\}\}\} b \approx f_{\omega 2+1}(n)\) *Multiexplosion \(= a \{\{\{2\}\}\} b \approx f_{\omega 2+2}(n)\) *Powerexplosion \(= a \{\{\{3\}\}\} b \approx f_{\omega 2+3}(n)\) *Explodotetration \(= a \{\{\{4\}\}\} b \approx f_{\omega 2+4}(n)\) *Detonation \(= \{a,b,1,4\} \approx f_{\omega 3}(n)\) *Pentonation \(= \{a,b,1,5\} \approx f_{\omega 4}(n)\) *Hexonation \(= \{a,b,1,6\} \approx f_{\omega 5}(n)\) *Heptonation \(= \{a,b,1,7\} \approx f_{\omega 6}(n)\) *Octonation \(= \{a,b,1,8\} \approx f_{\omega 7}(n)\) *Ennonation \(= \{a,b,1,9\} \approx f_{\omega 8}(n)\) *Deconation \(= \{a,b,1,10\} \approx f_{\omega 9}(n)\) *CG function \(= cg(n) \approx f_{\omega^2}(n)\) *Exploding Tree Function \(= E(n) \approx f_{\omega^2}(n)\) *Megotion \(= \{a,b,1,1,2\} \approx f_{\omega^2+1}(n)\) *BOX_M~ function \(= \{a,b,1,1,2\} \approx f_{\omega^2+1}(n)\) *Multimegotion \(= \{a,b,2,1,2\} \approx f_{\omega^2+2}(n)\) *Powermegotion \(= \{a,b,3,1,2\} \approx f_{\omega^2+3}(n)\) *Megotetration \(= \{a,b,4,1,2\} \approx f_{\omega^2+4}(n)\) *Megoexpansion \(= \{a,b,1,2,2\} \approx f_{\omega^2+\omega+1}(n)\) *Multimegoexpansion \(= \{a,b,2,2,2\} \approx f_{\omega^2+\omega+2}(n)\) *Powermegoexpansion \(= \{a,b,3,2,2\} \approx f_{\omega^2+\omega+3}(n)\) *Megoexpandotetration \(= \{a,b,4,2,2\} \approx f_{\omega^2+\omega+4}(n)\) *Megoexplosion \(= \{a,b,1,3,2\} \approx f_{\omega^2+\omega 2}(n)\) *Megodetonation \(= \{a,b,1,4,2\} \approx f_{\omega^2+\omega 3}(n)\) *Gigotion \(= \{a,b,1,1,3\} \approx f_{(\omega^2) 2+1}(n)\) *Gigoexpansion \(= \{a,b,1,2,3\} \approx f_{(\omega^2) 2+\omega}(n)\) *Gigoexplosion \(= \{a,b,1,3,3\} \approx f_{(\omega^2) 2+\omega 2}(n)\) *Gigodetonation \(= \{a,b,1,4,3\} \approx f_{(\omega^2) 2+\omega 3}(n)\) *Terotion \(= \{a,b,1,1,4\} \approx f_{(\omega^2) 3+1}(n)\) *Petotion \(= \{a,b,1,1,5\} \approx f_{(\omega^2) 4+1}(n)\) *Hatotion \(= \{a,b,1,1,6\} \approx f_{(\omega^2) 5+1}(n)\) *Hepotion \(= \{a,b,1,1,7\} \approx f_{(\omega^2) 6+1}(n)\) *Ocotion \(= \{a,b,1,1,8\} \approx f_{(\omega^2) 7+1}(n)\) *Nanotion \(= \{a,b,1,1,9\} \approx f_{(\omega^2) 8+1}(n)\) *Uzotion \(= \{a,b,1,1,10\} \approx f_{(\omega^2) 9+1}(n)\) *Uuotion \(= \{a,b,1,1,11\} \approx f_{(\omega^2) 10+1}(n)\) *Udotion \(= \{a,b,1,1,12\} \approx f_{(\omega^2) 11+1}(n)\) *Utotion \(= \{a,b,1,1,13\} \approx f_{(\omega^2) 12+1}(n)\) *Ueotion \(= \{a,b,1,1,14\} \approx f_{(\omega^2) 13+1}(n)\) *Upotion \(= \{a,b,1,1,15\} \approx f_{(\omega^2) 14+1}(n)\) *Uhotion \(= \{a,b,1,1,16\} \approx f_{(\omega^2) 15+1}(n)\) *Uaotion \(= \{a,b,1,1,17\} \approx f_{(\omega^2) 16+1}(n)\) *Uootion \(= \{a,b,1,1,18\} \approx f_{(\omega^2) 17+1}(n)\) *Unotion \(= \{a,b,1,1,19\} \approx f_{(\omega^2) 18+1}(n)\) *Dzotion \(= \{a,b,1,1,20\} \approx f_{(\omega^2) 19+1}(n)\) *Tzotion \(= \{a,b,1,1,30\} \approx f_{(\omega^2) 29+1}(n)\) *Uzzotion \(= \{a,b,1,1,100\} \approx f_{(\omega^2) 99+1}(n)\) *Hurford's C function \(= C(n) \approx f_{\omega^3 + \omega}(n)\) From \(f_{\omega^\omega}(n)\) to \(f_{\varepsilon_0}(n)\) *Linear array notation \(= \{a,b (1) 2\} \approx f_{\omega^\omega}(n)\) (limit) *Extended hyper-E notation \(xE\# \approx f_{\omega^\omega}(n)\) (limit) *n(k) function \(\approx f_{\omega^\omega}(n)\) *Taro's multivariable Ackermann function \(\approx f_{\omega^\omega}(n)\) *Planar array notation \(= \{a,b (2) 2\} \approx f_{\omega^{\omega^2}}(n)\) (limit) *Extended array notation (dimensional) \(= \{a,b (0,1) 2\} \approx f_{\omega^{\omega^\omega}}(n)\) *BEAF (superdimensional) \(= \{a,b (\underbrace{0,0\ldots0,0,1}_{n}) 2\} \approx f_{\omega^{\omega^{\omega^\omega}}}(n)\) (limit) From \(f_{\varepsilon_0}(n)\) to \(f_{\Gamma_0}(n)\) Starting from here, the totality of these functions is not provable in Peano arithmetic. *BEAF (tetrational) \(= {^ba} \& n \approx f_{\varepsilon_0}(n)\) (limit) *Cascading-E notation \(E\text{^} \approx f_{\varepsilon_0}(n)\) (limit) *Goodstein function \(= G(n) \approx f_{\varepsilon_0}(n)\) *Hydra(n) function \(\approx f_{\varepsilon_0}(n)\) *Nested Cascading-E Notation \(\approx f_{\varphi(\omega,0)}(n)\) From \(f_{\Gamma_0}(n)\) to \(f_{\omega^\text{CK}_1}(n)\) These functions and all those that follow cannot be proved total in arithmetical transfinite induction. *tree(n) function \(\approx f_{\theta(\Omega^\omega)}(n)\) *TREE(n) function \(> f_{\theta(\Omega^\omega)}(n)\) *BEAF (L-space) \(= \{a,b / 2\} \approx f_{\theta(\Omega^\Omega)}(n)\) (limit) *BEAF \(\approx f_{\theta((\Omega^\Omega)2)}(n)\) (estimated limit) *Bird's H(n) function \(\approx f_{\theta(\epsilon_{\Omega+1})}(n)\) *Bird's S(n) function \(\approx f_{\theta(\theta_1(\Omega))}(n)\) *Bird's U(n) function \(\approx f_{\theta(\Omega_\omega)}(n)\) *Bird's array notation \(\approx f_{\theta(\Omega_\omega)}(n)\) (limit) *SCG(n) function \(\geq f_{\psi_{\Omega_1}(\Omega_\omega)}(n)\) *BH(n) function \(\geq f_{\psi_0(\epsilon_{\Omega_\omega + 1})}(n)\) *Loader.c \(= D(n) \gg f_{\theta(\Omega_{\Omega_{\Omega_\cdots}})}(n) = f_{\psi(\psi_I(0))}(n)\) *:No notation has yet been devised to describe the ordinal for this function. Beyond \(f_{\omega^\text{CK}_1}(n)\) These functions are uncomputable, and cannot be evaluated by computer programs in finite time. *Rado's sigma function (and most of its variations) \(= \Sigma(n) \approx f_{\omega^\text{CK}_1}(n)\) *mth order sigma function \(= \Sigma_m(n) \approx f_{\omega^\text{CK}_m}(n)\) *Xi function \(= \Xi(n) \approx f_{\alpha}(n)\), where \(\alpha\) is \(\epsilon_0^{CK}\), the first fixed point of \(\alpha \mapsto \omega^\text{CK}_{\alpha}\). *1st Aarex function \(\approx f_{\alpha+\omega}(n)\) *2nd Aarex function \(\approx f_{\alpha+\omega^2}(n)\) *3rd Aarex function \(\approx f_{\alpha+\omega^{\omega^2}}(n)\) *4th Aarex function \(\approx f_{\alpha+\omega^{\omega^\omega}}(n)\) *5th Aarex function \(\approx f_{\beta+\omega^{\omega^\omega}}(n)\), where \(\beta\) is \(\phi(\omega,1)^{CK}\). *Rayo's function \(\gg f_{\omega_\alpha^\text{CK}}(n)\) *: The ordinal describing Rayo(n) is large enough that no notation is yet known for it. Other *Fusible margin function *: The growth rate of the \(m_1(x)\) function is an unsolved problem. *Hyperfactorial array notation *: HAN is new and still in development; its growth rate has not yet been conclusively evaluated. *Slow-growing hierarchy, Hardy hierarchy, Fast-growing hierarchy *: These three hierarchies can extend indefinitely, as long as ordinals and their fundamental hierarchies can be defined. *Iota function Category:Functions Category:Lists